Minimum Variance Unbiased Maximum Likelihood Estimation of the
Extreme Value Index
Roger Gay
New results for ratios of extremes from distributions with
a regularly varying tail are presented. Deriving from independence results
for certain functions of order statistics, 'consecutive' ratios of extremes
are shown to be independent as well as non-distribution specific. They have
tractable distributions related to beta distributions. The minimum variance
unbiased maximum likelihood estimator has the form of Hill's estimator. It
achieves the Cramer-Rao minimum variance bound and is a function of a sufficient
statistic. For small sample sizes the form of the moment generating function
of the estimator shows it has a gamma distribution.
Keywords: Tail-index, Minimum variance unbiased, Maximum likelihood,
Asymptotically normal
